The maximum-margin classifier is posed as a convex program with a quadratic objective and linear inequality constraints, whose Lagrangian dual — obtained by stationarity in the weight vector and the intercept — is a simpler quadratic program in the multipliers subject only to non-negativity and a linear equality constraint. Because strong duality holds, the KKT conditions characterize the solution, and complementary slackness forces the multiplier of every point strictly outside the margin to vanish, so the weight vector is a signed linear combination of only those training points lying on the margin — the support vectors. The concept belongs to statistical learning theory and convex optimization as applied to supervised classification, and it explains why margin-based classifiers are stable under perturbations of non-boundary data, in contrast to density-estimation classifiers whose parameters depend on every observation.
The maximum-margin classifier is posed as a convex program with a quadratic objective and linear inequality constraints, whose Lagrangian dual — obtained by stationarity in the weight vector and the …